Accessing large global charge via the ϵ-expansion

We compute the lowest operator dimension ∆(J; D) at large global charge J in the O(2) Wilson-Fisher model in D = 4 − ϵ dimensions, to leading order in both 1/J and ϵ. While the effective field theory approach of [1] could only determine ∆(J; 3) as a series expansion in 1/J up to an undetermined constant in front of each term, this time we try to determine the coefficient in front of J3/2 in the ϵ-expansion. The final result for ∆(J; D) in the (resummed) ϵ-expansion, valid when J ≫ 1/ϵ ≫ 1, turns out to be$$ \Delta \left(J;D\right)=\left[\frac{2\left(D-1\right)}{3\left(D-2\right)}{\left(\frac{9\left(D-2\right)\pi }{5D}\right)}^{\frac{D}{2\left(D-1\right)}}{\left[\frac{5\Gamma \left(\frac{D}{2}\right)}{24{\pi}^2}\right]}^{\frac{1}{D-1}}{\epsilon}^{\frac{D-1}{2\left(D-1\right)}}\right]\times {J}^{\frac{D}{D-1}}+O\left({J}^{\frac{D-2}{D-1}}\right) $$ Δ J D = 2 D − 1 3 D − 2 9 D − 2 π 5 D D 2 D − 1 5 Γ D 2 24 π 2 1 D − 1 ϵ D − 1 2 D − 1 × J D D − 1 + O J D − 2 D − 1 where next-to-leading order onwards were not computed here due to technical cumbersomeness, despite there are no fundamental difficulties. We also compare the result at ϵ = 1,$$ \Delta (J)=0.293\times {J}^{3/2}+\cdots $$ Δ J = 0.293 × J 3 / 2 + ⋯ to the actual data from the Monte-Carlo simulation in three dimensions [2], and the discrepancy of the coefficient 0.293 from the numerics turned out to be 13%. Additionally, we also find a crossover of ∆(J; D) from ∆(J) ∝ $$ {J}^{\frac{D}{D-1}} $$ J D D − 1 to ∆(J) ∝ J, at around J ∼ 1/ϵ, as one decreases J while fixing ϵ (or vice versa), reflecting the fact that there are no interacting fixed-point at ϵ = 0. Based on this behaviour, we propose an interesting double-scaling limit which fixes λ ≡ Jϵ, suitable for probing the region of the crossover. I will give ∆(J; D) to next-to-leading order in perturbation theory, either in 1/λ or in λ, valid when λ ≫ 1 and λ ≪ 1, respectively.